The force exerted by an orthodontic wire is directly proportional to its cross-sectional area, which depends on the square of the diameter (A = πd²/4). Doubling the diameter increases the area by 4 times. Additionally, the moment of inertia, which influences wire stiffness, is proportional to the diameter raised to the fourth power (I ∝ d⁴). Therefore, doubling the diameter increases stiffness — and thus force exerted — by 2⁴ = 16 times. This explains why doubling the diameter results in a 16-fold increase in force exerted by the wire.

Reference: Graber's Orthodontics, 7th Edition.